geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, the compound of three cubes is a

uniform polyhedron compound In geometry, a uniform polyhedron compound is a polyhedral compound whose constituents are identical (although possibly enantiomorphous) uniform polyhedra, in an arrangement that is also uniform, i.e. the symmetry group of the compound acts tran ...

formed from three

cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only r ...

s arranged with

octahedral symmetry A regular octahedron has 24 rotational (or orientation-preserving) symmetries, and 48 symmetries altogether. These include transformations that combine a reflection and a rotation. A cube has the same set of symmetries, since it is the polyhedr ...

. It has been depicted in works by

Max Brückner Johannes Max Brückner (5 August 1860 – 1 November 1934) was a German geometer, known for his collection of polyhedral models. Education and career Brückner was born in Hartau, in the Kingdom of Saxony, a town that is now part of Zittau, ...

and

M.C. Escher Maurits Cornelis Escher (; 17 June 1898 – 27 March 1972) was a Dutch graphic artist who made Mathematics and art, mathematically inspired woodcuts, lithography, lithographs, and mezzotints. Despite wide popular interest, Escher was for ...

History

This compound appears in

's book ''Vielecke und Vielflache'' (1900), and in the

lithograph Lithography () is a planographic method of printing originally based on the immiscibility of oil and water. The printing is from a stone (lithographic limestone) or a metal plate with a smooth surface. It was invented in 1796 by the German a ...

print ''

Waterfall A waterfall is a point in a river or stream where water flows over a vertical drop or a series of steep drops. Waterfalls also occur where meltwater drops over the edge of a tabular iceberg or ice shelf. Waterfalls can be formed in several wa ...

'' (1961) by

, who learned of it from Brückner's book. Its dual, the

compound of three octahedra In mathematics, the compound of three octahedra or octahedron 3-compound is a polyhedral compound formed from three regular octahedra, all sharing a common center but rotated with respect to each other. Although appearing earlier in the mathemati ...

, forms the central image in an earlier Escher

woodcut Woodcut is a relief printing technique in printmaking. An artist carves an image into the surface of a block of wood—typically with gouges—leaving the printing parts level with the surface while removing the non-printing parts. Areas that ...

, ''

Stars A star is an astronomical object comprising a luminous spheroid of plasma held together by its gravity. The nearest star to Earth is the Sun. Many other stars are visible to the naked eye at night, but their immense distances from Earth ma ...

''. In the 15th-century manuscript ''

De quinque corporibus regularibus ''De quinque corporibus regularibus'' (sometimes called ''Libellus de quinque corporibus regularibus'') is a book on the geometry of polyhedra written in the 1480s or early 1490s by Italian painter and mathematician Piero della Francesca. It is ...

'',

Piero della Francesca Piero della Francesca (, also , ; – 12 October 1492), originally named Piero di Benedetto, was an Italian painter of the Early Renaissance. To contemporaries he was also known as a mathematician and geometer. Nowadays Piero della Francesca i ...

includes a drawing of an octahedron circumscribed around a cube, with eight of the cube edges lying in the octahedron's eight faces. Three cubes inscribed in this way within a single octahedron would form the compound of three cubes, but della Francesca does not depict the compound.

Construction and coordinates

This compound can be constructed by superimposing three identical cubes, and then rotating each by 45 degrees about a separate axis (that passes through the centres of two opposite faces).

Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...

for the vertices of this compound can be chosen as all the

permutation In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or proc ...

s of

(0,\pm 1,\pm\sqrt)

References

{{reflist, refs= {{citation, first=Max, last=Brückner, author-link=Max Brückner, title=Vielecke und Vielflache, Theorie und Geschichte, location=Leipzig, publisher=B.G. Teubner, year=1900, a
Plate 23
} {{citation, url=http://www.georgehart.com/virtual-polyhedra/piero.html, last=Hart, first=George W., authorlink=George W. Hart, contribution=Piero della Francesca's Polyhedra, title=Virtual Polyhedra, year=1998. {{citation, contribution=Max Brücknerʼs Wunderkammer of Paper Polyhedra, last=Hart, first=George W., authorlink=George W. Hart, title=Bridges 2019 Conference Proceedings, url=http://archive.bridgesmathart.org/2019/bridges2019-59.pdf, pages=59–66 {{citation, last=Verheyen, first=Hugo F., contribution=Chapter 4: Classification of the finite compounds of cubes, doi=10.1007/978-1-4612-4074-7_5, isbn=0-8176-3661-7, location=Boston, mr=1363715, pages=95–159, publisher=Birkhäuser, series=Design Science Collection, title=Symmetry Orbits, year=1996; see in particula
p. 136
Polyhedral compounds